In the preceding sections of this chapter, we presented our mathematical framework as it applies to the pricing of American options on a single stock paying a continuous dividend yield. The same approach can be used to treat more general problems which have a similar structure. One example is index options. Here the underlying is not a single stock, but an index, such as S&P 500 or FTSE 100 or Nikkei 225. Another example is foreign exchange options, i.e., options to buy or sell foreign currency at a specified exchange rate. In this case, the role of the continuous dividend yield is played by the foreign interest rate. We perform calculations for several options of these types.
0 200 400 600 800 1000 1200 0 100 200 300 400 500 600 Index Value Option Price, $ American European
Early exercise premium
(a) American vs. European options
0 0.2 0.4 0.6 0.8 1 1.2 380 400 420 440 460 480 500 520
Time to expiration, years
Index value
(b) Optimal exercise price
Figure 7.6: American S&P 100 put with r = 0.05, σ = 0.25 and K = $520, 1 year from expiry.
tion traded on the Chicago Board of Options Exchange and the options written on it are American-style; therefore they are suitable for our analysis. We present numerical results for put options only, neglecting any dividend streams paid out by the constituents of the index. On May 9, 2002, the S&P 100 closed at 531.69, and we take a slightly in-the-money put with strike price K = $520. We take the interest rate equal to the yield on a 10-year treasury bond, or 5%, and the volatility equal to the value of the market volatility index, or 25%.3 We plot the results in Figure 7.6(a) for the value of the option and early exercise
premium and Figure 7.6(b) for the optimal exercise boundary. The calculated value of the index, at which early exercise is optimal one year before expiration is 390.40.
Our two final examples deal with call options to buy two different currencies for US dollars. We choose the euro and the Japanese yen. The fundamental difference is in the interest rate that can be earned on these currencies: in the euro zone, it is about 3.3% and in Japan, 0.8%. Here we take one-week short rates, the analog of which in the US is currently 1.78%. Identifyingq =rf for the foreign interest rate, we see that the dollar-euro
call satisfies r < rf, while the dollar-yen call satisfies r > rf, so we have both cases here.
We take the volatilities to be 9.6% for the euro and 9% for the yen, which is consistent with the numbers in the current financial press.
Results appear in Figures 7.7 and 7.8 on pages 102 and 103. The corresponding critical exercise exchange rates one year from expiration of the option are 0.816 dollars per euro and 1.90 dollars per 100 yen, respectively. Note that the early exercise premium for the yen option is extremely small, while that for the euro option is in line with the previous model calculations.
3On May 9, 2002, the yield on a 10-year T-bond was 5.187%, and the volatility index VIX closed at
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0
0.5 1
Exchange Rate, USD/EUR
Option Price, $
American
European
Early exercise premium
(a) American vs. European options
0 0.2 0.4 0.6 0.8 1 1.2 0.92 0.94 0.96 0.98 1 1.02 1.04
Time to expiry, years
Exchange rate, $/Euro
(b) Optimal exercise rate
Figure 7.7: American option to buy Euro at the exchange rate K = 0.92 ($ per ¿) within 1 year; r= 0.018,rf = 0.033,σ = 0.096.
0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2
Exchange Rate, USD/100 JPY
Option Price, $
American
European
Early exercise premium
(a) American vs. European options
0 0.2 0.4 0.6 0.8 1 1.2 1.8 1.82 1.84 1.86 1.88 1.9 1.92
Time to expiry, years
Exchange rate, $/100 yen
(b) Optimal exercise rate
Figure 7.8: American option to buy Japanese yen at the exchange rate K = 0.8 ($ per ¥ 100) within 1 year; r= 0.018,rf = 0.008,σ= 0.09.
Chapter 8
Concluding Remarks
I hope that posterity will judge me kindly, not only as to the things which I have explained, but also to those which I have intentionally omitted so as to leave to others the pleasure of discovery. (Rene Descartes, 15961650)
8.1 Summary and conclusions
We have presented a general framework of a front-fixing method for moving boundary prob- lems for linear parabolic equations, based on the Chebyshev expansion of solutions. Highly accurate and rapidly convergent for smooth problems, this method becomes a very powerful and robust tool in the presence of singularities, when most other techniques lose accuracy, require analytical start-off expressions (which may in many cases not exist), or, otherwise, fail altogether. The intuitively simple construction based on smooth approximations of sin- gular initial data, introduced in this text, allows us to use the accurate spectral framework, and keep control over the convergence and the error in numerical solutions. We have proved, in the most general linear setting, the convergence of numerical solutions of approximated problems to the true solution of the original problem, as the accuracy of the approximation of the singular initial data by smooth functions increases, and have given convincing numer- ical evidence of the same behavior in the moving boundary case. We have provided simple modifications of the general method, relying on analytic continuation and prior integration, which allow for considerable gain in computing time for certain practical problems. We have incorporated domain decomposition into the general framework, which enhances accuracy in localized settings. The oxygen diffusion test problem was used to illustrate the power of our method in the classical framework: the theoretical predictions of quadratic convergence as
smooth data approximates singular data were verified perfectly. The approach proposed in this work, which is applicable in its full generality, compares very favorably to the existing techniques, including those which, relying on integral equation formulations, were designed specifically for the treatment of problems of a type analogous to this test problem.
In the investigation of the American option problem, we re-established an important parity result, which allowed us to formulate, classify and solve this problem from the point of view of the theory and numerical analysis of partial differential equations. Even though this approach is gaining popularity, our work is one of the very few systematic efforts to analyze this challenging finance problem from an applied mathematics angle. Our numerical method produced meaningful results both for the classical setting of one asset and for more practical examples, including index options and foreign currency options. We find the success of our general method in the treatment of these quite singular problems very encouraging indeed.